Sliding modes in Banach spaces. (English) Zbl 1031.93050

A concept of a generalized solution for autonomous semilinear differential equations (with state-discontinuous right-hand side) in a reflexive Banach space is defined. This definition makes sense for a control system with discontinuous feedback control law and describes its motion on a sliding surface. Under suitable assumptions, it is shown that there exists only one generalized solution which is viable on the sliding surface and coincides with the one arising from the application of the equivalent control method [Yu. V. Orlov and V. I. Utkin, Automatica 23, 753-757 (1987; Zbl 0661.93036)]. The results obtained are applied to an exponential stabilization problem.


93B12 Variable structure systems
93C25 Control/observation systems in abstract spaces
34A60 Ordinary differential inclusions
34G25 Evolution inclusions
93D15 Stabilization of systems by feedback


Zbl 0661.93036